The FCC-ee offers powerful opportunities for direct or indirect evidence for physics beyond the standard model, via a combination of high-precision measurements and searches for forbidden and rare processes and feebly coupled particles. A key element of FCC-ee physics program is the measurement of the Z lineshape from a total of 5 x 10(12) Z bosons and a beam-energy calibration with relative uncertainty of 10(-6). With this exceptionally large event sample, five orders of magnitude larger than that accumulated during the whole LEP1 operation at the Z pole, the defining parameters-m(Z), Gamma(Z), N-nu, sin(2)theta(eff)(W), alpha S(m(Z)(2)), and alpha QED(m(Z)(2))-can be extracted with a leap in accuracy of up to two orders of magnitude with respect to the current state of the art. The ultimate goal that experimental and theory systematic errors match the statistical accuracy (4 keV on the Z mass and width, 3x10(-6) on sin(2)theta(eff)(W), a relative 3x10(-5) on aQED, and less than 0.0001 on aS) leads to highly demanding requirements on collider operation, beam instrumentation, detector design, computing facilities, theoretical calculations, and Monte Carlo event generators. Such precise measurements also call for innovative analysis methods, which require a joint effort and understanding between theorists, experimenters, and accelerator teams.